Pollock's conjectures are closely related conjectures in additive number theory.[1] They were first stated in 1850 by Sir Frederick Pollock,[1][2] better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers.

Statement of the conjectures

The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence A000797 in the OEIS) of 241 terms, with 343867 being almost certainly the last such number.[3]

  • Pollock octahedral numbers conjecture: Every positive integer is the sum of at most 7 octahedral numbers.

This conjecture has been proven for all but finitely many positive integers.[4]

  • Pollock cube numbers conjecture: Every positive integer is the sum of at most 9 cube numbers.

The cube numbers case was established from 1909 to 1912 by Wieferich[5] and A. J. Kempner.[6]

This conjecture was confirmed as true in 2023.[7]

  • Pollock icosahedral numbers conjecture: Every positive integer is the sum of at most 13 icosahedral numbers.
  • Pollock dodecahedral numbers conjecture: Every positive integer is the sum of at most 21 dodecahedral numbers.

References

  1. 1 2 Dickson, L. E. (June 7, 2005). History of the Theory of Numbers, Vol. II: Diophantine Analysis. Dover. pp. 22–23. ISBN 0-486-44233-0.
  2. ↑ Frederick Pollock (1850). "On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders". Abstracts of the Papers Communicated to the Royal Society of London. 5: 922–924. JSTOR 111069.
  3. ↑ Weisstein, Eric W. "Pollock's Conjecture". MathWorld.
  4. ↑ Elessar Brady, Zarathustra (2016). "Sums of seven octahedral numbers". Journal of the London Mathematical Society. Second Series. 93 (1): 244–272. arXiv:1509.04316. doi:10.1112/jlms/jdv061. MR 3455791. S2CID 206364502.
  5. ↑ Wieferich, Arthur (1909). "Beweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt". Mathematische Annalen (in German). 66 (1): 95–101. doi:10.1007/BF01450913. S2CID 121386035.
  6. ↑ Kempner, Aubrey (1912). "Bemerkungen zum Waringschen Problem". Mathematische Annalen (in German). 72 (3): 387–399. doi:10.1007/BF01456723. S2CID 120101223.
  7. ↑ Kureš, Miroslav (2023-10-27). "A Proof of Pollock's Conjecture on Centered Nonagonal Numbers". The Mathematical Intelligencer. doi:10.1007/s00283-023-10307-0. ISSN 0343-6993.


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